Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces |
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| Authors: | Ke Ang Fu Li Xin Zhang |
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| Affiliation: | (1) Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China;(2) School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310035, P. R. China |
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| Abstract: | Let {X, X n ; n ≥ 1} be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with covariance operator Σ. Set S n = X 1 + X 2 + ... + X n , n ≥ 1. We prove that, for b > −1, holds if EX = 0, and E‖X‖2(log ‖X‖)3b∨(b+4) < ∞, where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator Σ, and σ 2 denotes the largest eigenvalue of Σ. Project supported by National Natural Science Foundation of China (No. 10771192; 70871103) |
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| Keywords: | the law of the logarithm moment convergence tail probability strong approximation |
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